
// Make newform 2960.2.a.r in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2960_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_2960_2_a_r();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_2960_2_a_r();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [2, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-3, 0, 1]];
    Rf_basisdens := [1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_2960_a();"
function MakeCharacter_2960_a()
    N := 2960;
    order := 1;
    char_gens := [2591, 741, 1777, 2481];
    v := [1, 1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_2960_a_Hecke(Kf)
    return MakeCharacter_2960_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 0, 0], [0, -1, 0], [-1, 0, 0], [0, 1, 0], [1, 0, 1], [0, -2, 0], [-2, 2, -2], [-1, 1, -1], [2, 0, -2], [-2, 0, 0], [1, 3, 1], [-1, 0, 0], [-5, 2, 1], [-2, 0, -2], [0, 1, 0], [-3, -2, 3], [5, -5, 1], [-4, 0, 2], [-3, -1, 1], [5, 2, 1], [-9, 2, 1], [1, -5, 1], [6, -3, 6], [-4, 4, -2], [4, 4, 2], [5, -4, 5], [-4, 2, 0], [-7, -1, 1], [-8, 4, -6], [-8, -2, 0], [-8, -1, -4], [-9, 5, -9], [-2, 4, -4], [0, -4, -4], [-11, 4, -3], [6, -6, 6], [-5, 2, -3], [-12, -2, 4], [-6, 2, -2], [1, -6, -1], [-1, 3, -1], [3, -4, -1], [7, -3, -1], [-2, 0, -8], [-1, -14, 5], [7, -1, 3], [-9, 2, 7], [0, -7, -4], [-6, -2, 2], [-1, -6, 1], [-2, -8, 4], [17, -3, -3], [-2, 0, 4], [5, 1, 5], [6, -6, -2], [-6, 11, -2], [12, 2, 0], [5, 2, 1], [6, 2, -2], [0, -4, -2], [-10, -6, 6], [-2, 0, 4], [-2, -9, 2], [13, -5, -3], [6, 0, 4], [2, 4, 0], [-21, -1, 3], [3, 6, -3], [-14, 0, -6], [8, 2, 4], [4, 4, -2], [7, 8, -5], [-17, -3, 7], [-7, 14, 3], [9, -4, 1], [4, 12, 0], [-12, 12, -10], [7, -6, -11], [14, -12, -4], [0, -12, 10], [25, 4, 1], [-4, -4, -2], [3, 3, -5], [13, -6, 3], [-7, 3, 5], [10, -7, 6], [0, -8, 2], [-2, 8, -4], [-2, 8, -4], [-2, -4, -6], [20, -10, 12], [-3, -7, -7], [6, -6, 2], [-4, 8, 8], [-5, 1, -9], [0, -12, 12], [17, -10, 15], [9, -8, -3], [-18, 4, -2], [-4, 0, -2], [-4, 4, -12], [-8, -10, 8], [-2, 4, -2], [-4, 8, -6], [7, -18, 7], [-6, 2, 2], [-4, -4, 4], [-17, -2, 5], [35, 4, -1], [-12, -14, 8], [-4, 8, -16], [-17, 14, 1], [-3, 2, 7], [11, 2, 3], [15, 17, -5], [-11, -12, 5], [-32, -6, 0], [-12, 8, -8], [-12, 12, 10], [-3, 14, -11], [0, 8, -2], [-9, -2, -7], [13, 2, 7], [18, 8, -6], [-6, 18, -2], [-8, -4, 2], [-4, 8, 10], [25, -8, 5], [-4, 0, 0], [-5, -6, -3], [27, -6, -5], [14, 5, 6], [-1, -16, 11], [-34, 8, -12], [-15, -2, -13], [22, -12, -12], [3, 18, -7], [-8, 17, -4], [-2, -6, -10], [14, -8, -4], [-25, -2, -9], [-7, -26, 7], [-1, 5, 3], [8, -2, -8], [8, 4, -2], [0, 20, 0], [-10, 0, -8], [-12, 12, 10], [-7, 7, -3], [21, -9, 13], [-6, 24, 0], [16, -18, 8], [-18, 14, 14], [20, -13, -4], [14, -2, 14], [5, 13, -3], [-27, 9, -19], [10, -16, 20], [-31, 2, -9], [12, -2, -12], [-14, 6, -14], [-35, 6, -9], [12, -2, 0], [0, -24, 4], [4, -4, -6], [28, 19, -8], [1, 9, -11], [6, -16, 12], [-5, 12, -1], [-4, 28, -10], [21, -19, 17], [2, -8, -4], [15, -23, -5], [19, -22, 9], [25, -14, -11], [4, -16, 10], [5, -29, 9], [12, -6, -8], [-8, 17, -8], [26, 8, 0], [-12, 8, 8], [-3, 1, 21], [26, 14, -2], [-28, 12, -22], [-7, 11, -3], [-19, 22, -1], [-15, -14, 7], [-18, 20, -2], [-34, -4, -4], [9, 0, 13], [16, -14, 12], [-11, 11, -15], [-21, 23, 7], [33, -10, -1], [6, 15, -14], [-7, -6, 15], [-4, 12, -2], [30, -10, 18], [3, 14, 9], [-32, 10, -4], [40, 12, -6], [5, -2, 1], [10, -12, 16], [17, -4, -7], [-18, -14, 2], [-22, 22, 6], [-8, 0, 8], [-3, 11, 13], [16, 12, 2], [-29, 6, -21], [-36, 2, 0], [-10, -4, -12], [-16, 0, 12], [7, 17, -13], [-19, 19, -7], [-31, 4, -15], [-30, 12, -10], [50, 8, -4], [-36, -2, 0], [-14, -12, 8], [-27, 0, -7], [-17, -30, 11], [-27, 20, -11], [-18, 8, 18], [22, 13, -2], [36, -12, 10], [26, -16, -4], [-15, 4, 13], [-22, 23, 2], [-21, 19, 3], [15, 14, 1], [25, 8, -7], [45, 4, -7], [-7, 22, 3], [-30, -7, 6], [-16, -5, -4], [10, 8, 4], [-12, 14, 12], [13, 5, -3], [11, 11, 3], [-8, 0, -24], [-59, 7, 1], [-20, 13, 0], [18, 20, 4], [14, -8, -4], [-53, -5, 3], [12, 18, -16], [23, 9, -13], [-1, 14, -9], [-14, -2, -18], [-10, -26, 2], [28, -6, 0], [-1, 13, 7], [32, 4, -2], [2, 32, -16], [3, 2, 11], [1, 6, 11], [41, -17, 5], [-19, 18, -13], [-6, 8, 4], [-4, -36, 8], [4, -20, 4], [-2, 16, -8], [-18, 12, -8], [6, 14, -22], [-37, 8, -13], [-1, -8, 15], [-40, -12, 10], [14, 3, -2], [6, 6, 10], [-32, 12, -18], [30, -2, 6], [20, 14, 4], [-3, -23, 13], [-26, 20, 12], [19, -19, 7], [17, -1, 5], [-12, 0, -6], [-17, 18, -11], [-19, 9, -15], [22, -29, 6], [5, -23, 17], [-15, -17, 21], [17, -10, -9], [28, -2, -4], [5, 12, -15], [12, 18, 0], [10, -12, 24], [-23, -11, -7], [2, -8, 20], [36, -16, 22], [2, 20, -22], [-31, 18, -1], [13, 14, -3], [41, -22, 15], [19, -6, -15], [39, 12, -5], [-10, -36, 16], [-15, -11, 5], [26, -13, -14], [-10, -32, 16], [-7, 6, -13], [1, -8, -11], [48, -16, 8], [3, -15, -9], [-48, -10, 0], [-2, -24, 6], [-6, 12, -32], [-49, -10, 11], [16, 10, 4], [10, -19, -14], [-37, -16, -5], [31, 24, -17], [8, 13, -4], [16, 4, 20], [-52, 16, 2], [21, -26, -3], [23, -23, 27], [-3, -2, -25], [14, -28, -4], [-17, 33, 3], [31, -6, 21], [14, 12, -16], [-37, -27, 7], [-47, 18, 3], [-16, -12, 22], [14, -2, -6], [-14, -22, -6], [-38, 28, 14], [-6, -8, -12], [-59, 6, -1], [32, -4, 18], [-31, 29, -3], [52, -4, -12], [11, -11, -21], [-31, 7, -23], [-55, 6, 11], [43, -2, -7], [8, -16, -6], [27, 9, -1], [13, -34, 23], [7, -2, 1], [-30, -4, 12], [1, -1, -11], [-26, 8, 4], [-17, 23, 3], [-43, -2, -13], [14, 41, -14], [-11, 7, 13], [-33, -2, 9], [22, -22, -6], [-1, -2, -3], [48, -4, -2], [52, 10, 0], [16, -10, 16], [54, 0, 0], [31, -7, -17], [37, -21, 13], [-9, -6, 13], [-12, 2, -16], [8, 30, -16], [13, 38, -21], [-36, 14, -16], [41, 17, -15], [-6, 7, 6], [14, -8, -4], [-4, -6, -16], [0, -10, 20], [-82, 8, 0], [-11, 5, 21], [11, -9, 15], [-13, -7, 3], [-60, 18, 0], [33, -3, 1], [-3, 10, -1], [22, -18, 2], [7, 18, -21], [-18, 24, -28], [-6, -12, -20], [1, 6, -21], [48, -8, 2], [-58, 14, -14], [24, -12, 12], [18, -20, 20], [-17, 17, 11], [52, -17, -8], [55, 2, -5], [22, 40, -8], [6, 40, -26], [-40, 18, -36], [-65, 18, 9], [26, -16, -20], [-59, 1, -3], [8, 38, -16], [5, -2, -23], [-10, 20, 4], [-1, 1, -1], [5, 6, 3], [17, -20, 17], [38, 8, 4], [7, -36, 3], [2, -40, 20], [4, -12, 24], [22, 16, 8], [26, -4, 16], [23, 22, -21], [30, -16, -20], [6, 0, -8], [41, 15, 9], [1, -8, 1], [71, -2, 9], [19, -2, 17], [32, -18, 16], [-25, 37, -25], [-8, -20, -18], [20, 14, 12], [-80, 0, 8], [20, -25, 8], [10, 40, -12], [-52, 28, 0], [0, -8, -34], [-6, 48, -16], [-69, -23, 7], [19, -12, -29], [6, -44, 20], [27, -10, 1], [3, -27, -1], [21, 22, 7], [34, -22, 14], [-41, 7, -13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2960_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2960_2_a_r();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_2960_2_a_r(:prec:=3)
    chi := MakeCharacter_2960_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_2960_2_a_r();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_2960_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2960_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, -4, 1, 1]>,<7,R![2, -4, -1, 1]>,<13,R![-16, -16, 2, 1]>],Snew);
    return Vf;
end function;